Initial-boundary value problems for linear PDEs th

arXiv:nlin/0210058v1[nlin.SI]24Oct2002INITIAL-BOUNDARYVALUEPROBLEMSFORLINEARPDEs:THEANALYTICITYAPPROACHA.Degasperis1,2,§,S.V.Manakov3,§andP.M.Santini1,2,§1DipartimentodiFisica,Universit`adiRoma”LaSapienza”Piazz.leAldoMoro2,I-00185Roma,Italy2IstitutoNazionalediFisicaNucleare,SezionediRomaP.leAldoMoro2,I-00185Roma,Italy3LandauInstituteforTheoreticalPhysics,Moscow,Russia§e-mail:[e-mail],[e-mail],[e-mail]-knownthatthemaindifficultiesassociatedwiththestudyofinitial-boundaryvalueproblemsforlinearPDEsisgivenbythepresenceofunknownboundaryvaluesinanymethodofsolution.Todealeffi-cientlywiththisdifficulty,wehaverecentlyproposedtwoalternative(butinterrelated)methodsinFourierspace:theAnalitycityapproachandtheEliminationbyRestrictionapproach.InthisworkwepresenttheAnalyt-icityapproachandweillustrateitspowerinstudyingthewell-posednessofinitial-boundaryvalueproblemsforsecondandthirdorderevolution-aryPDEs,andinconstructingtheirsolution.Wealsoshowtheconnec-tionbetweentheAnalyticityapproachandtheEliminationbyRestrictionapproachinthestudyoftheDiricheletandNeumannproblemsfortheSchr¨odingerequationinthen-dimensionalquadrant.1IntroductionItiswell-knownthatthemaindifficultiesassociatedwithInitial-BoundaryValue(IBV)problemsforlinearPDEsofthetypeL(▽,∂∂t)u(x,t)=f(x,t),u(x,0)=u0(x),x∈V⊂Rn,t0,(1)where▽=(∂∂x1,··,∂∂xn),Lisaconstantcoefficientspartialdifferentialoperator,u(x,t)istheunknownfield,f(x,t)isagivenforcingandu0(x)isthegiveninitialcondition,withDirichelet,orNeumann,orRobin,ormixed,orperiodic1boundarydataon∂V,isgivenbythepresenceofunknownBoundaryValues(BVs)inanymethodofsolution.Todealefficientlywiththisdifficulty,wehaverecentlyproposedtwoalternative(butinterrelated)methodsinFourierspace:theAnalyticityapproachandtheEliminationbyRestrictionapproach.Thefirststep,commontobothmethods,consistsinrewritingthePDE(1),definedinaspace-timedomainD,inthecorrespondingFourierspace,usingtheGreen’sformula.ThePDEinFourierspacetakestheformofalinearrelationamongtheFourierTransforms(FTs)ofthesolution,oftheinitialconditionandofasetofBVs,onlyasubsetofwhichisgivenapriori.ThisrelationisalwayssupplementedbystronganalyticityrequirementsonalltheFTsinvolved,consequenceofthegeometricpropertiesofthespace-timedomainD.Thesecondstepiswherethetwomethodsseparate;oncetheproblemisformulatedinFourierspace,weproposethefollowingtwoalternativestrategies.i)TheAnalyticityapproach,whichconsistsinusingsystematicallytheanalyt-icitypropertiesofalltheFTsinvolvedintheaboverelation,toderiveasystemoflinearequationswhichallowsonetoexpresstheunknownBVsintermsoftheknownones,andthereforetosolvetheproblem.ii)TheEliminationbyRestriction(EbR)approach,whichconsists,instead,inapplyingtotheabovelinearrelationinFourierspaceasuitableannihilationoperator,whicheliminatesalltheunknownBVs,generatinganewtransform,well-suitedtothespecificIBVproblemunderscrutiny.Theinversionofthisnewtransform(ifitexists)leadstothesolution.TheAnalyticityapproachisinspiredbyFokas’recentdiscoveryoftheglobalrelation,obtainedfirstwithinthex−ttransformapproach[1]andmorerecentlyusingdifferentialforms[2].Theuseoftheglobalrelationtostudythewell-posednessandsolveIBVproblemsisillustrated,forinstance,in[3],[4],[5].In[5],inparticular,generalresultsonthewell-posednessofIBVproblemsfordispersive1+1dimensionalequationsofarbitraryorderareannounced.Ourmaincontributiontothemethodconsists,afterformulatingtheIBVprobleminFourierspaceusingGreen’sformula,inimposingsystematicallytheanalyticitypropertiesofalltheFouriertransformsinvolvedintheproblem,toderiveacascadeofanalyticityconstraintswhichallowonetoexpresstheunknownBVsintermsoftheknownones,andthereforetosolvetheproblem.Inparticular,Fokas’globalrelationappears,inthemethodologywepropose,asa“zeroresiduecondition”fortheFTofthesolution.TheAnalyticityapproachintheformweproposeisveryelementaryand,aboveall,hasthegreatconceptualadvantagetooriginatefromasingleguidingprinciple:theanalyticityofalltheFouriertransformsinvolvedintheproblem.ItisthetypeofapproachthatcanbeeasilytaughtinelementaryUniversitycourses,combiningnicelystandardPDEtheorytools,liketheGreen’sformulaandtheFouriertransform,withelementarynotionsinComplexFunctionsthe-ory.TheessentialaspectsoftheAnalyticityapproachwerefirstpresentedbytheauthorsattheWorkshop“Boundaryvalueproblems”inCambridge,De-2cember2001,insidetheprogramme:“IntegrableSystems”.Themethodinitsfinalformispresentedforthefirsttimeinthispaper,illustratedonthestudyofIBVproblemsofvarioustype(Dirichelet,Neumann,mixed,periodic)forsomesecondandthirdorderclassicalPDEsoftheMathematicalPhysics:theSchr¨odinger,theheatandthelinearKorteweg-deVriesequations.Alsoitscon-nectionswiththeEbRapproachareillustratedinthiswork,ontheparticularexampleoftheSchr¨odingerequationinthen-dimensionalquadrant.AgoodaccountoftheEbRapproachisgiveninsteadin[6].Adifferentapproach,validforsemicompactdomains,hasbeenrecentlypresentedin[7].AgeneralreviewofthebasicspectralmethodsofsolutionofIBVproblemsforlinearandsolitonPDEsispresentedin[8].§2isdevotedtothepresentationoftheAnaly